4. Find orthonormal vectors 91, 92, and q3 such that 9₁ and 92 span the column space of the matrix 1 2 43 A = 2 -3 -2 4 Which of the four fundamental subspaces contains 93? Solve Ax = b, where b = (1, 2, 4)T by least squares.

Find a basis for the column space of 4 1 3 0 - -31 3 2 1 4 -1 A -3 -2 -1 - -4 2 6 -4 -2 -8 3 Basis }. ={ 9.

Find a and that solve the vector equation: Suppose the above equation is expressed as Ax = b with x = coeffieints of the equations in one matrix form as (Ab)= Rows: 2 Columns: 3 b) c) d) 21 a (²) + ^ ( ;) = (²1) B α= B = Complete the missing column of the following matrix derived by doing row operation of the augmented matrix in a): 1 -2 (²). 0 1 Express the

Consider the following set of vectors. 3 1 ~-~--~-- = 3 = Let V₁, V₂, and v3 be (column) vectors in R3 and let A be the 3 x 3 matrix V₁ Consider the following equation. V₁, V₂, and v3 are linearly dependent if and only if the homogeneous linear system with augmented matrix [A10] has a nontrivial solution. C₁ + C₂ V3 = 3 + C₂ 3 = 0 0 0 V1 V2 V3 with these vectors as its columns. Then Solve for C₁, C₂, and c3. If a nontrivial solution exists, state it or state the general solution in terms of the parameter t. (If only the trivial solution exists, enter the trivial solution (C₁, C₂, C3} = {0, 0, 0}.) (C₁, C₂, C3} = . Determine if the vectors V₁, V₂, and v3 are linearly independent. O The set of vectors is linearly dependent. O The set of vectors is linearly independent.