2. Let B-[vi..... Vn] be an orthonormal basis of R". Prove that the matrix P (vilv2l...Vn) is ort...
7. Claim: Let A be an (n × n) (square) matrix. ·Claim: If A s invertible and AT = A-1 , then the columns of A form an orthonormal basis for R . Claim: If the columns of A form an orthogonal basis for Rn, then A is invertible and A A-1 . Claim: If the columns of A form an orthonormal basis for R", then A is invertible and AT= A-1 . Claim: If the columns of A form...
5. Let (a) (2 marks) Find all eigenvalues of A (b) (4 marks) Find an orthonormal basis for each eigenspace of A (you may find an orthonormal basis by inspection or use the Gram-Schmidt algorithm on each eigenspace) (c) (2 marks) Deduce that A is orthogonally diagonalizable. Write down an orthogonal matrix P and a diagonal matrix D such that D P-AP. (d) (1 mark) Use the fact that P is an orthogonal matrix to find P-1 (e) (2 marks)...
11. Let Vi,b, . . . , vn, and u be vectors in Rm. Determine whether or not the following two statements mean the same thing. Explain your answer. Statement l: u is in Sp(vi, V2,. . . , Vn). Statement 2: If A-Vi U2 Vis an m x n matrix, there exists a vector i in R" such that A
2.4. Let V be a vector space and let vi,V,..., Vn be a basis in V. For x Prove that (x, y) defines an inner product in V
Let P be an orthogonal matrix. (a) Prove that detP = 1 or detP = −1. (b)If detP = −1, show that I+P has no inverse. Hint: PT(I+P)=(I+P)T. (which T means transpose)
, Vn be vectors in IR" with (vi,. .., v vn is aso 2. Let vi..., linearly dependent. Show that , 3. Let T' R3 -IR3 be defined by T(2:1, 2:2, 23) (27 + 22, 2x2 + x3, xs), (a) Find the standard matrix representing T (b) Determine if T is one-to-one. (c) Determine if T is onto.
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Please explain why each is true or false (examples would be helpful thank you!) 2. Let vi,.. . ,vn E R", and let A - [vi. .. vn]. Suppose A-b has no solutions for some b. Circle all true statements b) (vi,..., Vn) is linearly independent c) Span(vi,... , vn) f R" d) Span(v1, . . . ,%) = Rk where k 〈 m e) Span(v1, . . . , vn) has dimension k where k < m f) Ax0m...
Problem 5. Let V and W be vector spaces, and suppose that B (vi, ..., Vn) is a basis of V a) Prove that for every function f : B → W, there exists a linear transformation T: V → W such that T(v;)-f(7) for all vEB (b) Prove that for any two linear transformations S : V → W and T : V → W, if S(6) = T(6) for all ï, B, then S = T (c) Prove...
Problem 6* (Optional). Suppose ej,..., en is an orthonormal basis of V and v, ...,Vn are vectors in V such that lle; - v, 1 < 1 h for each j. Prove that V1, ..., Vn is a basis of V. In other words, if you perturb an orthonormal basis slightly, you still have a basis.