Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary. Р W= 1 S (2) -2p+2q=3s -p=3s + 2r Rewrite the system of equations in the form Ax = 0. Ax= P 2 498830 - 10 S

3 13 Given A = - 4 - 12 - 4 observe that the first column plus twice the third column equals the second column, Find a nontrivial solution of Ax = 0 without -3 - 15 9- performing row operations. [Hint: Write Ax = 0 as a vector equation.] X =

Express each of the following vectors in R² as linear combinations of the vectors [4] and (a) (b) (c) (d) 16 11 -36 17 -12 15 -20 = || = || - 4 Y + Y + -4 + Y + + + + A 4 -

The vectors -3 -3 -9 w 3= 12 20 + k 15 are linearly independent if and only if k +

Plz solve both

This

Please include easy and clear explanation for each solving step

One of the following vectors lies in span{ 2+ x + x², 2x2} (A) 2+ 2x + 7x? (B) 2 + 2x + 5x? (C) 2 + 2x + 6x? (D) 2 + 2.x + 8x? (E) 2 + 2x + 9x2