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MAT211_PracticeMidterm1[1]

MAT211_PracticeMidterm1[1] - following matrix invertible a...

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1 Example (3.3-27) Determine whether the following vectors form a basis of R 4 (1,1,1,1), (1,-1,1,-1),(1,2,4,8),(1,-2,4,-8) Exercise 1.2-30 Find the polynomial of degree 3 whose graph passes through the points (0,1),(1,0),(-1,0), (2,-15) Find the inverse of the rotation matrix. cos(a) -sin(a) sin(a) cos(a) Let T be a clockwise rotation in R 2 by π /2 followed by an orthogonal projection onto the y axis. 1. Find the matrix of T. 2. Determine whether T is invertible 3. Find im(T) and ker(T Find the inverse of the matrix. Interpret your result geometrically. a b b -a For the matrix A below, find all the 2x2 matrices X that satisfy the equation A . X=I 2. 1 2 3 5
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2 3.1-23 Describe the image and kernel of the reflexion about the line y=x/3 in R 2 . Compute the dimensions of the kernel and the image. (2.4-31)For which values of the constants a, b and c is the
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Unformatted text preview: following matrix invertible? a b-a c-b-c 3.2-46 ±ind a basis of the kernel and image of the matrix. Determine the dimensions of the kernel and image. Determine the rank. Justify your answers. 1 2 3 5 1 4 6 Example (3.3-31) Let V be the subspace of R 4 de²ned by the equation x 1-x 2 + 2x 3 + 4 x 4 = 0 ±ind a linear transformation T from R to R 4 such that ker(T)={0}, im(T)=V. Describe T by its matrix. Give an example of a 5 x 4 matrix A with dim(ker A)=3. Compute dim(im A). Consider the vectors of R 5 , (1,1,0,0,0), (0,0,0,2,2), (1,1,0,1,1), (0,0,1,0,0). Compute the dimension of the subspace V of R spanned by those vectors. Are they linearly independent? Is (1,2,0,0,0) a linear combination of those vectors?...
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