Math 304

Does the set {( 1 3 1), (2, 7, -3), (4 8 -7)} span R3?

5.1 Put into row echelon form. If there are 3 rows with variables it does.

Find the best approximation of x^2 + 1 on [−1, 1] by a linear function y = ax + b with respect to the inner product <f|g> = R [1 −1] f(x)g(x) dx.

9.4

determinant using row reduction

row reduce until everything below the diagonal is a 0. Then multiply the diagonal.

trace(A)

sum of numbers on the diagonal

matrix multiplication size rules

the first matrix must have the same number of columns as the second matrix has rows. The number of rows of the resulting matrix equals the number of rows of the first matrix, and the number of columns of the resulting matrix equals the number of columns of the second matrix.

Find distance from the point (1, 0, 0) to the set of solutions of the system x_1 + 2x_2 + x_3 = 0 x_1 + 3x_2 − x_3 = 0

10.1 Step 1: Solve for the solution to the system of equations Step 2: Use the distance formula to find the distance D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2) Answer: sqrt(30t^2 - 10t + 1)

Check that the matrix (1/2 1/2 1/2 1/2 −1/2 −1/2 1/2 1/2 −1/2 1/2 1/2 −1/2 1/2 −1/2 1/2 −1/2) is orthogonal. Does it have a real eigenvalue? Is it orientation preserving?

10.2 1. Check if matrix is orthogonal - A matrix is orthogonal is Q^T = Q^-1 or QQ^T = I - Find Transpose of the matrix - interchange the rows and columns - Multiply transpose by original matrix - If the result is the identity matrix, then it is orthogonal Answer: yes 2. Solve for eigenvalues - det(A - λI) = 0, solve for λ - to find determinant of matrix do gaussian elimination until all entries below the main diagonal are 0 Answer: no 3. if determinant of the matrix is positive it is orientation preserving. If the determinant is negative then it is orientation reversing. Answer:

Use the Gram-Schmidt orthogonalization to find an orthogonal basis for the subspace spanned by (2 −1 3 1) , (−3 0 −1 4) in R 4 .

10.3 1. u_1 = v_1, u_2 = v_2 - ((v_2 • u_1) / (u_1 • u_1))*u_1 or v_2 - proj_(u_1) (v_2) Answer:{(u_1), (u_2)} {(2, -1, 3, 1), (-7/3, -1/3, 0, 13/3)}

Use the Gram-Schmidt orthogonalization to find an orthogonal basis for the subspace spanned by (4 −1 −2 2), (8 −1 4 0), (−1 2 0 −2)

10.3 1. u_1 = v_1, u_2 = v_2 - ((v_2 • u_1) / (u_1 • u_1))*u_1 or v_2 - proj_(u_1) (v_2) u_3 = v_3 - ((v_3 • u_2)/(u_2 • u_2))*u_2 - ((v_3 • u_1)/(u_1 • u_1))*u_1 or v3 - proj_(u1)(v_3) - proj_u2(v3) Answer: {(4, -1, -2, 2), (4, 0, 6, -2), (3/5, 8/5, -4/5, -6/5)}

Consider the space C([−1, 1]) with the inner product <f | g> = R [1, −1] f(x)g(x)x^2 dx. Use Gram-Schmidt orthogonalization to the sequence 1, x, x2 , x3 .

10.4

Find the inverse of the matrix using the adjoint matrix.

4.1 Adjoint = transpose of cofactor matrix 1. find cofactor matrix - find the minor of each element of the matrix and alternate signs. 2. Transpose this 3. Get the determinant of the original matrix plug into formula

Solve the system using Cramer's Rule.

4.2 x1 = detx1/det x2 = detx2/det x3 = detx3/det when solving for x1 determinant replace x1 row with solutions row...repeat for x2 and x3

Determine whether the following sets form subspaces of R 3 .

4.3 The condition must be satisfied for (0, 0, 0), (x1 + y1, x2 + y2, x3 + y3), and c(x1, x2, x3)

Which of the following subsets are subspaces of the space of 2 × 2 matrices under the usual matrix operations.

4.4 Check zero matrix, adding another matrix, and multiplying matrix by a constant

Which of the following sets of vectors are linearly independent in R 3 ?

4.5 If there are no free variables when it is reduced, the set is independent

Find bases of the column space, row space, and null space of the matrices

5.2 1. Column space Do row echelon form to eliminate any rows that are completely zeros. However many rows have numbers n are the number of vectors that are in the column space. The first n columns from the original matrix is the basis of the column space 2. row space The nonzero rows in row echelon form 3. null space Set matrix equal to zero and solve

Find coordinates of ~v in the ordered basis B in the following cases.

5.3 Put B into matrix form and set it equal to v, then row reduce until its the identity matrix. The answer is the numbers on the right side.

Find the transition matrix from B to C.

5.4 Set matrices equal to each other. Solve until left side is identity matrix. Right side is the answer

Suppose that ( 1 2 −1) are coordinates of a vector with respect to the basis B of the previous problem. What are its coordinates with respect to the basis C?

5.5 Multiply matrix c by coordinates of the vector

Determine which of the following functions are linear transformations. 1. f : R 3 → R 3 given by h(x1, x2, x3) = (x2, x3, x1); 2. f : R 2×2 → R given by f ( a b c d) = ad−bc; 3. f : P3 → R given by f(a_3x^3 + a_2x^2 + a_1x + a_0) = a_3 +a_2 +a_1 +a_0; 4. f : R n×n → R, where f(A) is the trace of A.

6.1 Function T is a linear transformation if T(a + b) = T(a) + T(b) T(ca) = cT(a)

Find the matrix of the linear transformation f : R 3 → R 2 given by f(x1, x2, x3) = (x1 - x3, x2)

6.2 Just convert the right hand side to matrix form

Find the standard matrix representation (i.e., the representation in the standard bases) for the following linear operators: (a) The linear operator R 2 −→ R 2 that rotates each vector ~x in R 2 by 45◦ in the clockwise direction. (b) The linear operator R 2 −→ R 2 that doubles the length of ~x and then rotates it by 30◦ in the counterclockwise direction. (c) The operator R 3 −→ R 3 given by the formula L (x1 x2 x3) = (2x3 x2 + 3x1 2x1 − x3)

6.3

Let L : R 3 → R 3 be given by L ( x y z) = ( −4y − 13z −6x + 5y + 6z 2x − 2y − 3z ) What is the matrix of L with respect to the standard basis of R 3 ? What is the matrix of L with respect to the basis ((-1 -6 2), (3 4 -1), (-1 -3 1))?

6.4 The matrix with respect to R2, literally just put it in matrix form easy L in new basis = S^-1AS S is new basis A is basis in R3

Find the characteristic polynomial of each of the given matrices. (2 5 8 0 -1 9 0 0 5), (5 1 4 1 2 3 3 -1 1)

7.1 cp = det(A - xI) 2. subtract a constant from each value in the diagonal 3. find the determinant

Find all eigenvalues and bases of the corresponding eigenspaces for the following matrices. (3 4 121 4 -12 3 12 3 -4), (8 -21 3 -8)

7.2 1. Find eignevalues - det(A - xI) = 0 2. For each eigenvalue plug back into the matrix as x, find solution 3. find basis from there

Which of the following matrices are diagonalizable? (-18 40 -8 18) (-3 3 -1 2 2 4 6 -3 4)

7.3 1. Find characteristic polynomial cp = det(A - xI) 2. Solve for eigenvalues cp = 0 3. If there are two different eigenvalues then there are two different eigenvectors. In order for it to be diagonizable it needs to have same number of real eigenvalues as the length of its row/column

In each of the following, factor the matrix A into a product SDS−1 , where D is diagonal. A = (5 6 -2 -2) A = (2 -8 1 -4) A = (1 0 0 -2 1 3 1 1 -1)

7.4 1. Find characteristic polynomial and solve for eigenvalues det(A - xI) = 0 2. Plug each eigenvalue back into the characteristic matrix and set the matrix equal to zero. 3. Solve for eigenvector 4. Repeat for all eigenvalues found 5. Take S as matrix of eigenvectors combined and D as identity matrix with eigenvalues on the diagonal 6. Invert S to get S^-1. To invert set matrix equal to identity matrix and use row reduction to reduce until the left side is the identity matrix. When doing e^D only do the diagonal entires of the matrix, leave the rest as they were

Compute e^A for A= ( 1 1 1 -1 -1 -1 1 1 1)

7.5 1. e^A = Se^D*S^-1 1. Find characteristic polynomial and solve for eigenvalues det(A - xI) = 0 2. Plug each eigenvalue back into the characteristic matrix and set the matrix equal to zero. 3. Solve for eigenvector 4. Repeat for all eigenvalues found 5. Take S as matrix of eigenvectors combined and D as identity matrix with eigenvalues on the diagonal 6. Invert S to get S^-1. To invert set matrix equal to identity matrix and use row reduction to reduce until the left side is the identity matrix.

Find the point on the line y = 2x that is closest to the point (5, 2).

8.1 1. Determine equation of perpendicular line - find perpendicular slope - do y - y1 = m(x - x1) to find equation 2. solve the two equations Answer: (9/5, 18/5)

Find the angle between the vectors (1, 1, 1, 1) and (1, 2, −1, 2) in R 4

8.2 Use formula cos(theta) = (u • v)/(|u||v|) Answer: arccos(2/sqrt(10))

Which of the following sets of vectors form an orthonormal basis of R 2 ? a) {(1 0) , (0 1)} b) {(3/5 4/5) , (5/13 12/13)} c) {(√ 3/2 1/2) , (−1/2 √ 3/2)}

8.3 1. Check if the set is orthogonal (dot product of vectors = 0) 2. Check if the set is orthonormal (magnitude of each vector = 1) Answer: a and c

Let S be the subspace of R 3 spanned by the vector (1, −1, 1)>. Find a basis of S ⊥.

8.4 1. the upside down T means orthogonal complement x1 = x2 - x3 (x1, x2, x3) = (x2-x3, x2, x3) = x2(1, 1, 0) + x3(-1, 0, 1) Answer: {(1, 1, 0), (-1. 0, 1)}

Find projection in R 4 (with the usual dot product) of the vector (1 −1 0 1) onto the span of the vectors (1 2 1 2) and (1 1 1 1).

9.1 1. Orthogonal projection is given by Ac. A^t*A*c = A^T * x c = (c_1, c_2) 2. A = (1 1 2 1 1 1 2 1) x = (1 -1 0 1) 2. Solve for (n x n)c = # 3. solve matrix for c1 and c2 4. Find Ac Answer: (1/2 0 1/2 0)

Find the least square solution of the system −x1 + x2 = 10 2x1 + x2 = 5 x1 − 2x2 = 20

9.2 1. The system is currently Ax = b We have to solve A^T * Ax = A^t b for x1 and x2 Answer: x1 = 19/7, x2 = -26/7

Find the best least square fit by a linear function y = ax + b of the data x = -1 0 1 2 y = 0 1 3 9

9.3 1. find least square solution of -a + b = 0 b = 1 a + b = 3 2a + b = 9 1. A = -1 1 0 1 1 1 1 1) Solve using formula A^T * Ax = A^t * b b = (0, 1, 3, 9) Answer: ax + b = 2.9x + 1.8