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 X txi,x2, x3, x4 be a set of 4 linearly independent vectors in R6. Let A be a 6 x 6 matrix of rank 5. Then the dimension of Span(A Ax2, Ax3, Ax4) (a) is equal to 4 since 5 4. (b) must be either 3 or 4. (c) must be 1, 2, or 3. (d) is equal to 5, since A has rank 5. (e) can be any number between 0 and 4 inclusive. (5 points) Suppose A is a m x n matrix. Which of the following is equal to the rank of A? (VI) (a) The dimension of V where V the column space of AT (b) The dimension of the null space of AT (c) The number of free variables in the system of equations Ax 30. (d) The dimension of V where V is the null space of A (e) None of the above.

Explanation / Answer

Rank of any matrix is the dimension of the vector space spanned by its columns.

This is same to the dimension of the space spanned by its rows.

Now it is given that A has a rank of 5 and it is a set of 4 linearly vectors in R^3.

Since 5>4, therefore the dimension of span is equal to 4

a) is the correct option.

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