\") Justify your answers to the following with brief explanations (a) True or fa

Question

") Justify your answers to the following with brief explanations (a) True or false? i. If there columns of A are linearly independent, then Ax = b has exactly one solution for every b ii. A 9 × 10 matrix never has linearly independent columns. iii. The column space of a 2 × 2 matrix is the same as its row space. iv. The column space of a 2 x 2 matrix has the same dimension as its row space. (b) Let A be a 64x17 matrix with rank 13. How many independent vectors satisfy Ax=0? How many independent vectors satisfy ATy = 0?

Explanation / Answer

Question A

1) Statement 1 is true. Since columns of A are linearly independent, the columns of A span Rn. Thus A is invertible, the statement is TRUE.

2) In a 9 x 10 matirx, there are 9 rows and 10 columns. These 10 columns lie in R9. Since R9 has dimension 9, we can't have more that 9 linearly independent variables in R10. This the statement is FALSE.

3) The statement is FALSE because the row space of matrix A is esentially the column space of AT.

4) Statement is TRUE. As the row space and column space has same dimension for a 2x2 matrix, it is called rank.

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