11. Prove that the identity vector in any vector space is unique. (Hint: use contradiction) 12....
12. Find bases for Nul A and Col A. (8pts) WA 5 -3 1 1 1 - 1 2 22 5 0 - 8 - 2 4 -4 8 3 - 2
Question 4 2 pts Determine whether the vector u is in the column space of the matrix A and whether it is the null space of A. 1 0 3 1 -2 1 - 4 U = 3 3 0 4 - 1 3 6 Not in Col A in Nul A In Col A, not in Nul A Not in ColA, not in Nul A In Col A and in Nul A Question 5 1 pts 1 co 2...
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. 2 3 8-11 A=1-6-6-12 18 4 -3 -20 23 A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A 1 2 02 A=177-21 351~1013-3 3 4 -6 12 3 3 -9 15
Question 3 (a) Use the defining equation for eigenvalues and eigenvectors to prove that if matrix A has the unique eigenvalue a, then the matrix A-al has 0 as an eigenvalue (b) Show that if matrix B has the eigenvalue then the matri B has 2 as an eigenvalue. (c) Use the defining equation to show that if the matrix C is invertible, then C cannot have zero as an eigenvalue. (Hint: No eigenvector Xcan be the zero vector. So...
[ 2 4 -2 11 4. (20pts) Consider a matrix A = 3 7 -8 6 and corresponding Col A & Nul A. -2 -5 7 3 Col A is a subspace of Rk and Nul A is a subspace of R'. |(1) Find k and one nonzero-vector in Col A. | (2) Find 1 and one nonzero-vector in Nul A.
if whoever answers this could be detailed with explanations please! 1 -3 -2 -5 -14 -6-3-8 -21 2 8.(15 pts. total) M= is row equivalent to -2 6 1 4 5 -9 -2-7 -14 + (1-3 0-1 0 0 0 2 7 2 Pivels L 0 0 0 0 0 0 0 0 0 0 (a) Find k and/ so that Nul Mc R and Col Mc R (b) Without calculations, list rank M and dim Nul M. (c) Find...
1. For the matrix A given below, find col(A) and Nul(A). Also determine if the given vector is in the column space, null space, both or neither. A = -2 -5 1 3 3 11 1 7 8 -5 -19 -13 0 1 7 5 -171 5 1 -3 1 5 1
Find the bases for Col A and Nul A. and then state the dimension of these subspaces for the matrix A and an echelon form of A below. 1 3 8 -1 -3 1 3 8 -1 -3 2 7 200 -4 0 1 4 2 -3 - 12 - 36 2 13000 3 13 40 0 -11 000 Abasis for Col A is given by (Use a comma to separate vectors as needed.)
(3 points.) Prove that the following properties hold in any vector space: (1) (-с)и %— — (си). (i) —(-и) %— (iii) (0)(x)= 0 и.
3. In the following question, we are going to prove that ker(T) = { } if and only if T is one-to- one. (Writing prove is like writing a little essay, with some good logical connection between each sentence.) (a) Let T:V - W a linear transformation between two vector spaces. Suppose ker(T)={0}. Show that T is one-to-one. (Hint: proof by contradiction, by assuming both ker(T)=ð and T is not one-to-one. Now, apply definition of kernel and one-to-one, what is...