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Find the standard matrix representations for each of the following linear operators. (a) L is the linear operator that rotates each x in R2 by 45 in the clockwise direction. (b) L is the linear operator that reflects each x in R2 about the x1 axis and then rotates it 90 in the counterclockwise direction.

Example 4: Find a basis for the column space of the matrix Since the column space of A consists precisely of those vectors b such that A x = b is a solvable system, one way to determine a basis for CS(A) would be to first find the space of all vectors b such that A x = b is consistent, then constructing a basis for this space.

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EXAMPLE: Let A 1 23 510 15, u 2 3 1, b 2 10 and c 3 0. Then define a transformation T : R3 R2 by T x Ax. a. Find an x in R3 whose image under T is b. b. Is there more than one x under T whose image is b.

standard inner product on R3. 3. (a) Find the radius and the centre of the circular section of the sphere Iri =4 cut off by the plane r . (i —j — k)= 2 (b) Apply the Cayley-Hamilton theorem to find the inverse of the matrix. 5 -1 2 2- A = 3 1 0 1 1 1 (c) Let S and T be the linear transformations from R2 to R2 defined by

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Identify the domain and codomain of the matrix transformations given by the following matrices, and determine whether the transformations are ono-to-one and onto. c 2 (b) Columns vectors of the matrix and Y2 — and let T: R2 —Y IR2 be a linear transformation that maps el into and mpas Example 3 Let 4 10 23 —11 4 15 15 into Y2.

A description of how a determinant describes the geometric properties of a linear transformation. ... given color in $[0,1]$ is mapped to a point of the same color in ...