Question

20 2. Provide an example for each of the following- 2 (a) A nonzero vector in R3 that is orthogonal to -1 3 (b) A 3 x 3 matrix which is not invertible. (c) A set of vectors in R2 which is linearly independent but not orthogonal. (d) A nonzero subspace of R.

Answer 1

(a.) Let E IR² such that 1 a b is orthogonal to Two vectors in R² are said to be orthogonal iff their dot product is zero 2ab + 3C = 0 One solution for this equation is a = 2 c =-1 b=1, 2 [ is orthogonal to (6.) А 3x3 matrix is non invertible iff it's zero. determinant is Consider A = 1 0 - 2 3 4 6 2 J'AIE 1(6-6)-0(4-4) +(12-12) = 0 6 - A = IS A 3 X 3 O 0 2 3 l 4 6 2 non-invertible matrix (c.) Let vi [3] 12 = [3] E IR? Here vi & kV2 V KER. : ū,,v2 are linearly independent Also viouz ( Dot product) 1 x 1 + 2 x 3 = 8 to J and 12 dre not orthogonal

Let S be a subspace of R. We need to find a non-zero subspace of R. :. Assume st{o} let non-zero element of s. Now a be a For every *tk, Xa es V KER (& is a scalar) :: S has to be the entire set of real nos. S = R. :. The only non- zero subspace of R is R itself.