Linear Transformations Flashcards

Questions and Answers

Define Linear Transformation and its properties.

A mapping from Rn to Rm denoted by T is linear if T(u+v) = Tu + Tv, T(αv) = αT(v), T(0) = 0, and T(au + bv) = aT(u) + bT(v) for every u, v ∈ Rn.

What is the standard matrix A?

Te1, Te2 * (x1@x2) = Ax.

What is the set Rn?

The domain of T.

What is the set Rm?

The co-domain of T. Signup and view all the answers

How is T completely determined?

By what it does in the columns of the identity matrix. Signup and view all the answers

Define a projection matrix.

A symmetric matrix B such that B^2 = B, for example, B = v*v^T where v is a unit vector. Signup and view all the answers

Define the image of x.

Vector b in Rm for x in Rn. Signup and view all the answers

What is a shear transformation?

T(x) = Ax; a map of a coordinate space in which one coordinate is held fixed and the other coordinates are shifted. Signup and view all the answers

Linear systems are expressed in terms of what?

Vector equations (Ax=b) or linear combinations. Signup and view all the answers

Vector equations are viewed as a?

Mapping view. Signup and view all the answers

Describe the mapping view.

Matrix A acts on x, sending it to b, illustrating a dynamic relationship. Signup and view all the answers

Explain the view of the matrix equation A = (1&-1&2@2&1&1), x = (1@1@1), b = (0@0).

Matrix A represents a map of R^3 to R^2, taking x to b. Signup and view all the answers

How does Ax=b lead to a transformation?

It finds all vectors in Rn sent to b in Rm under a map. Signup and view all the answers

What is the precise solution set of Ax=b?

The set of all pre-images of b. Signup and view all the answers

In particular, N(A) is...

The set of all vectors in Rn that are sent to 0. Signup and view all the answers

Describe a matrix-vector multiplication.

Mapping of Rn to Rm represented by A of size m by n. Signup and view all the answers

What is the pre-image of a matrix A of size m by n?

x-vector. Signup and view all the answers

What is the image of x?

Ax or b. Signup and view all the answers

Describe the range of A and how we get it.

The set of all images Ax from Rn under A. Signup and view all the answers

What is the image of b = (1@2) for A = (1&-1&2@2&1&1), x = (1@0@0)?

(1@2). Signup and view all the answers

How do you find a pre-image?

Solve a linear system and determine a particular solution. Signup and view all the answers

Can you determine a particular solution of this matrix equation by inspection: A = (1&-1&2@2&1&1), x = (x@y@z), b = (2@4)?

(1@1@1). Signup and view all the answers

What is the pre-image of (2@4) for A = (1&-1&2@2&1&1), x = (x@y@z)?

(1@1@1) + N(A). Signup and view all the answers

Express N(A) as a span when A = (1&-1&2@2&1&1).

span {-1@1@1}. Signup and view all the answers

What is the range of A for A = (1&-1&2@2&1&1)?

R^2. Signup and view all the answers

Describe the first geometric example in linear transformations.

A projection from R^3 onto the xy plane represented by A. Signup and view all the answers

Describe the second geometric example in linear transformation.

Transformation examples vary but usually illustrate rotations, dilations, or projections. Signup and view all the answers

Study Notes

Linear Transformations Overview

A linear transformation maps vectors from Rn to Rm and is denoted as T.
Properties of linear transformations include:
- T(u + v) = Tu + Tv (additivity)
- T(αv) = αT(v) (homogeneity)
- T(0) = 0 (maps the zero vector to zero)
- T(au + bv) = aT(u) + bT(v) for scalars a, b.

Standard Matrix Representation

The standard matrix A is defined by the action of T on the standard basis vectors e1 and e2, represented as: Te1, Te2 * (x1@x2) = Ax.

Vector Spaces

Rn represents the domain of the linear transformation T.
Rm is the co-domain of T.

Determining T

T is completely determined by its effects on the columns of the identity matrix In.

Projection Matrix

A projection matrix B is symmetric and satisfies B^2 = B, exemplified by B = v*v^T, where v is a unit vector.

Image and Shear Transformations

The image of a vector x is the resulting vector b in Rm after applying T.
A shear transformation involves fixing one coordinate while shifting others: T(x) = Ax.

Linear Systems and Vector Equations

Linear systems can be expressed through vector equations (Ax = b).
Vector equations can be understood in a "mapping" context, where matrix A transforms vector x into vector b.

Mapping View

Viewing Ax = b as a mapping illustrates how matrix A acts on vector x to output vector b. This dynamic perspective emphasizes relationships between input and output vectors.

Matrix Equations and Transformations

A matrix equation can be interpreted as a transformation from Rn to Rm, mapping pre-images (x) in the domain to images (b) in the co-domain.
The null space, N(A), is the set of all vectors in Rn that are sent to the zero vector.

Range of A

The range of matrix A is the collection of all images Ax produced for vectors from Rn, representing the output space of the transformation.

Finding Pre-images and Solutions

To determine a pre-image, solve the linear system, find a particular solution, and add elements from the null space of A.

Geometric Interpretation

The first geometric example involves mapping from R^3 to R^3, where A = (1&0&0@0&1&0@0&0&0) projects onto the xy-plane, transforming coordinates (x,y,z) to (x,y,0).

Further Exploration

The second geometric example remains to be defined, inviting further inquiry into geometric implications of linear transformations.

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Description

Explore essential definitions and properties of linear transformations through this interactive flashcard quiz. Each card addresses key concepts, including the standard matrix representation and the linearity criteria. Perfect for anyone studying linear algebra.