For the following multiple choice questions, you may give one (and only one) ans

Question

For the following multiple choice questions, you may give one (and only one) answer. Select the single best option.

(5 points) Let Ps be the vector space of all polynomials of degree less than or equal to 3. Write D for the derivative, regarded as a linear transformation from P3 to P2 Then (IX) (a) The dimension of the kernel of D is 2 (b) The polynomial 2r 1 is in the kernel of D (c) The polynomial 2ars 1 is in the image of D (d) The kernel of D consists of all polynomials with zero constant term. (e) The rank of the matrix of D is 3. (5 points) The dimension of the null space of a 3 x 5 matrix (X) (a) Can be any number from two to five. (b) Can be any number from zero to two. (c) Must be two (d) Must be three (e) Can be any number from zero to three. Must be zero. (g) Can be any number from zero to five.

Explanation / Answer

1) Dimension of null space matrix of 3 x 5 matrix:

can be any number from zero to two.

2) rank of 5x3 matrix can be

anything from zero to five

3) det(A+B) = detA + detB

= 6 + 4 = 10

Option D

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