(in psychology) "the attribution of one's own ideas, feelings or attitudes to other people or to objects; esp: the externalization of blame, guilt, or responsibility as a defense against anxiety"1

— or — to paraphrase Dorland's Medical Dictionary: a mental defensive mechanism whereby a "repressed complex" is concealed by being assigned as the property of the world at large or as a property of someone besides the person who is engaging in the projection.

Psychoanalytic terms, including "projection" and "transference," as they apply to a therapeutic setting (and most specifically, a setting in which the analysand is a child) are discussed in brief at:


The page referenced discusses these terms in the context of "psychoanalytic play technique as devised by Melanie Klein." The author is psychoanalyst Chris Mawson.

Another interesting source for definitions on these matters would appear at the C.G. Jung Page. "Projection" is defined at:


and includes the comment that "transference" and "countertransference" are each terms used specifically to identify the type of projection that occurs commonly in a analyst-analysand relationship, refering respectively to projections by the analysand and by the analyst.

1 Webster's Ninth Collegiate Dictionary

In the unreal world that is psychotherapy, this is considered a good thing. Good, in the sense that once a patient/client starts projecting his beliefs and fears onto the therapist a clearer picture of what really is bothering the person comes to light. When the discussion moves to why the client feels the therapist is acting a certain way it is an opening to why the client is acting that way (or afraid to act a certain way).

The act or process of projection, although a defense mechanism, is fuel for the theraputic process. A standard way to tell when this process is occuring is when a client begins to comment on how the therapist is not "sharing of himself" and is closed off from the client (similar to transference).

In geometry, the transformation of points from one space to another.

One example application is cartography, where geographic features such as coastlines, rivers, political boundaries, etc., that lie on a sphere (okay, the earth isn't quite a sphere) must be rendered on two-dimensional plane (i.e., a map). Many kinds of projections have been invented for this task, each with its strengths and shortcomings (e.g., preserving land areas or preserving direction), which may depend on the size of the region being projected.

A simpler application is photography, where the three-dimensional world is transformed into a two-dimensional plane using the perspective projection. A line (called a line of sight is drawn between a special point (called the focal point and each point in the scene, and the intersection of these lines with a specially designated plane (termed the focal plane) forms an image.

An even simpler application is non-perspective architectural drawings such as blueprints. There is no focal point; lines of sight are all parallel and intersect the focal plane orthogonally. Thus, this is usually termed an othogonal projection.

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As you are doing the exercises try building a picture in your mind of what is going on. The picture does not have to be visual, it could be a rough understanding of which pins are set and how much resistance you are encountering from each pin. One way to foster this picture building is to try to remember your sensations and beliefs about a lock just before it opened. When a lock opens, don't think "that's over", think "what happened".

This exercise requires a lock that you find easy to pick. It will help you refine the visual skills you need to master lock picking. Pick the lok, and try to remember how the process felt. Rehearse in your mind how everything feels when the lock is picked properly. Basically you want to create a movie that records the process of picking the lock. Visualize the motion of your muscles as they apply the correct pressure and torque, and feel the resistance encountered by the pick. Now pickthe lock again trying to match your actions to the movie.

By repeating this exercise, you are learning how to formulate detailed commands for your muscles and how to interpret feedback from your senses. The mental rehearsal teaches you how to build a visual understanding of the lock and how to recognize the major steps of picking it.

In Computer Science, a projection function is one of the basic building blocks of primitive recursive functions. Given a k-dimensional array, it returns the ith element, i.e., pik: NkN where p(x1 ,..., xk) = xi. This is heavily used due to the way in which composition is defined for primitive recursive functions- to obtain the effect of (g o f), i.e., g(f(x)) it is necessary to use projection to extract the single argument,x- g=Cn[f,p11]. In effect the projection reduces k-dimensional space into one dimensional space.

In Linear Algebra, a projection is a linear map on a vector space V, π : V→V, which satisfies a further constraint: π2=π; that is, v ∈ V, π(π(v)) = π(v). It follows that any higher powers of π are also equivalent to π, by repeated application of the definition of a projection. The power notation arises from the fact that composition of linear maps is equivalent to multiplication of corresponding matrices- if a matrix P represents a projection π, the product PP = P2 represents π o π = π, so powers of P are simply P itself.

Two fairly obvious projections exist: the zero map and the identity map. This makes intuitive sense, as 02=0 and 12=1. In the context of maps, repeated application of the zero map is redundant once it has been applied once, and the identity map never changes the value operated on and hence may also be applied an arbitrary number of times to no effect. It turns out that all projections can be thought of as a particular combination of these two maps:

Theorem: For a vector space V and a linear map α:V→V ; α is a projection iff there are subspaces U,W of V such that

  • V= U W
  • α: u+w → u u∈U, w∈W.

Two proofs give very different insights into how this works. The first is self-contained, appealing only to the properties given or requested; whilst the second shows how a more general and hence powerful result of linear algebra, the primary decomposition theorem, can be applied to obtain an even more elegant proof.

Note first that the projections we identified earlier, the zero and identity maps, are rather degenerate cases of this theorem. For the zero map, W is V, whilst the other subspace is the trivial set {0} - Note that whilst this means that U and W have a common element, the zero vector, this is permitted by the definition of direct sum- there may be no other common vector, however. Thus any element of V is an element of W (itself) plus the zero vector from U, and gets mapped to that element of U; hence zero is yielded in every case. For the identity map, we take U to be the vector space V, and let W be {0}. Now every vector is mapped to itself (the thing in U). Thus this suggests a method of construction for a general projection.

Proof 1, forward implication
Suppose we have a linear map α: V→V such that α2=α. Take candidates for U and W as follows: U = Im α , W = Ker α .
We claim that V = U⊕W as desired.

Let v be an arbitrary element from V. Then obviously v= v + α(v) - α(v). Rearranging gives v= α(v) + (v-α(v)).
By definition of image, α(v) ∈ Im α = U. So we need to show that (v-α(v)) ∈ Ker α = W to obtain V = U+W.
The definition of Ker α is that it contains all elements of V such that when α is applied to them, the zero vector is obtained. So take α of (v-α(v)): = α(=v) - α(α(v)). Since α is a projection, this becomes α(v) - α(v) = 0 as desired.

We have V = U + W. We seek a direct sum; meaning that the expression of a v from V as u+w from U,W should be unique. This is equivalent to U∩W = {0}.

So consider x∈U∩W. Then x∈U=Im α, so there is a y∈V such that x=α(y). Further, x∈W=Ker α so α(x)=0. Yet α(x)=α(α(y)) = α(y) by property of projection = x. So x∈U∩W implies x=0: so U∩W ⊆ {0}. 0 is in U and W, so 0 is in U∩W, giving {0} ⊆ U∩W. So U∩W = {0}.

Hence V = U⊕W . We desired also that if v = u+w, then α(v)= u. However, α(v) = α(u+w) which by linearity gives α(v) = α(u) + α(w). Since u∈U=Im α, u=α(u') for some u' and by property of projection α(u) is therefore u, whilst w∈W=Ker α so by defintion α(w)=0. Hence α(v) = u + 0 = u . We are done.

Proof 1, backward implication
Suppose that V decomposes as a direct sum; i.e., if v∈V then v=u+w for unique u∈U, w∈W. We define α(v) = u, which by uniqueness is a well-defined function. It remains to show linearity and that α is a projection.

Let v=u+w and v'=u'+w' be elements of V. Now for scalars λ,μ from the field,
α(λv + μv') = α(λ(u+w) + μ(u'+w'))
=α( (λu + μu') + (λv + μv'))
=λu + μu' , Since U is a vector space over the field of V, λu + μu' ∈ U. So we can apply the definition of α
=λα(u) + μα(u')
So the definition of linearity is met. As α(α(v)) = α(u) = u = α(v), we also have that this map is a projection. We are done.

Introducing some more technology we can simplify this considerably (although some of the insight into the nuts and bolts of the process is lost.)

Proof 2
We have already discussed that the decomposition works for the identity and zero maps. So consider a projection π:V→V which is a projection but neither of these two maps. Then we can write the definition as π2 - π = 0. Hence π satisfies the polynomial p(t) = t2 -t. Since this is a degree two monic polynomial, it follows that the minimal polynomial of π is at most degree two.

Now P(t) = t(t-1). If the minimal polynomial has degree one rather than two, then it must be either m(t)=t or m(t)=(t-1), since any polynomial that is satisfied by a linear map has its minimal polynomial as a factor. But if m(t)=t, we have the zero map, and if m(t)=(t-1), we have the identity map. We assumed that neither of these applied; so the minimal polynomial cannot be of degree one. It is hence of degree two, and in fact must be p(t) = t(t-1) by uniqueness.

Now the primary decomposition theorem applies. mπ(t)=p(t)=q1(t)q2(t) with q(1)=(t-1), q2(t)=(t) coprime. So V = W1⊕W2 where the Wi are π invariant subspaces with minimal polynomial of π|Wi given by qi.
But then q1 indicates that π|W1 is the identity map, whilst q2 indicates that π|W2 is the zero map.
So given v=w1+w2, wi ∈ Wi, π(v) = π(w1 + w2) = π(w1) + π(w2) = id(w1) + zero(w2) = w1 + 0 = w1 . This completes the proof.

Pro*jec"tion (?), n. [L. projectio: cf. F. projection.]


The act of throwing or shooting forward.


A jutting out; also, a part jutting out, as of a building; an extension beyond something else.


The act of scheming or planning; also, that which is planned; contrivance; design; plan.


4. Persp.

The representation of something; delineation; plan; especially, the representation of any object on a perspective plane, or such a delineation as would result were the chief points of the object thrown forward upon the plane, each in the direction of a line drawn through it from a given point of sight, or central point; as, the projection of a sphere. The several kinds of projection differ according to the assumed point of sight and plane of projection in each.

5. Geog.

Any method of representing the surface of the earth upon a plane.

Conical projection, a mode of representing the sphere, the spherical surface being projected upon the surface of a cone tangent to the sphere, the point of sight being at the center of the sphere. -- Cylindric projection, a mode of representing the sphere, the spherical surface being projected upon the surface of a cylinder touching the sphere, the point of sight being at the center of the sphere. -- Globular, Gnomonic, Orthographic, projection,etc. See under Globular, Gnomonic, etc. -- Mercator's projection, a mode of representing the sphere in which the meridians are drawn parallel to each other, and the parallels of latitude are straight lines whose distance from each other increases with their distance from the equator, so that at all places the degrees of latitude and longitude have to each other the same ratio as on the sphere itself. -- Oblique projection, a projection made by parallel lines drawn from every point of a figure and meeting the plane of projection obliquely. -- Polar projection, a projection of the sphere in which the point of sight is at the center, and the plane of projection passes through one of the polar circles. -- Powder of projection Alchemy., a certain powder cast into a crucible or other vessel containing prepared metal or other matter which is to be thereby transmuted into gold. -- Projection of a point on a plane Descriptive Geom., the foot of a perpendicular to the plane drawn through the point. -- Projection of a straight line of a plane, the straight line of the plane connecting the feet of the perpendiculars let fall from the extremities of the given line.

Syn. -- See Protuberance.

<-- projectionist. one who operates a projector[2]; esp. one who is employed to operate a movie projector in a movie theater -->


© Webster 1913.

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