1. True or False: a). b) Every basis of R³ has 2 vectors The standard basis of P4, polynomials of degree less than or equal to 4, is {1,1, x², x³, x4¹} c). The matrix equation Ax = b is consistent if and only if b is in the col(A), the column space of A The set {x d) e) - If T: V→ spaces V, W then f). h) R² : 2+2 ≤ 1} is a vector subspace of R² W is a linear transformation between finite dimensional vector dim(V) = dim(N(T)) + dim(R(T)) A linear transformation is one to one if its nullspace contains 6 nonzero vectors. Every n x n symmetric matrix is orthogonally diagonalizable. An eigenvector is always nonzero.

let V and W be two vector spaces. A is a set of basis for V and B is a set of basis for W. a) Consider the given linear mapping T: V -> W which satisfies given equations for a1 a2 and a3 b) then find T(v) where v_a is the given matrix

7. Let V be a vector space and suppose B = {u, v} is a basis for V. Prove that B' = {ku, v} is also a basis for V for every nonzero scalar k. (Hint: You must prove that B'is linearly independent and that it spans V).

2. In a vector space V with inner product (,), the vectors u and v are orthogonal and ||u|| ||v|| = 5. Determine the value of (u + 3v, 2u - 4v). = 3 and

33. Prove that the row vectors of an n x n invertible matrix A form a basis for Rn.

1- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (-3,0,4), (5,-1,2), (1,1,3) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³ 4- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthogonal basis for R³, then obtain an orthonormal basis from them. Express (2,3,-4) as a linear combination of the orthonormal basis. 5- Find the coordinates of the vector u = (0,25, 1) with respect to the orthogonal basis v₁ = ( 3,4,0), V₂ = (−4, 3, 0 ) and v₂ = (−4,3,0) 6- Find the coordinates of the vector U = (7,5,2) with respect to the basis V₁ =…

18. Find a basis and dimension of the orthogonal complement to the span ((1, 2, 0, -1), (0, 1, 1, 0), (1, 1, −1, −1)).