15. Suppose T : R^2 R^3 is linear defined by T(x, y) =(x + 3y, x - y, x). Then
(i) 1
(ii) 2
(iii) 3
(iv) None
16. Suppose T : R^3 R^3 is linear and has an upper-triangular matrix with respect to the basis (1,0,0),(1,1,1),(1,1,2). Then, the orthonormal basis of R^3 with respect to which T has an upper-triangular matrix is...
(i) (1, 0, 0), (0, 1/(√2), 1/(√2)), (0, - 1/(√2), 1/(√2))
(ii) (1, 0, 0), (0, 1, 0), (0, 1/(√2), - 1/(√2)
(iii) (1, 0, 0), (0, - 1, 1), (0, 1, 1)
(iv) None
17. Which of the following defines an inner product
(i) <(x base 1, x base 2), y base 1, y base 2)>2x base 1 y base 1 +x base 2 y base 2 in R^2
(ii) <(x base 1, x base 2), y base 1, y base 2)>x base 1 y base 1 +2x base 2 y base 2 - 1 in R^2
(iii) <a base 1 + b base 1 x +c base 1 x^2, a base 2 +b base 2 x + c base 2 x^2 > = a base 1 b base 1 +a base 2 b base 2 +c base 1 c base 2 in P base 2
15)
rref of
Rank of T is 2
Correct option is
16)
Using Gram-Schmidt process
Let
Correct option is (i)
17)
For a finite-dimension polynomial vector
Space:
Option (iii) is the correct one
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