1. 1 1 1 01 0 10 0 0 0 0 0 1 1 0 0 Lo o o 0 1 1 X1 X2 X3 X4 X5 X6 1 1. 2 2 2 Problem 3: The 4 x 6 matrix A and its row-reduced echelon form are: A = 0 3 3 3 3 4 4. X1 X2 X3 X4 X5 X6 a. Fill out the following table of properties of the matrix: Dimension of the Dimension of Dimension of Is the transformation Is the transformation Rank Domain of A Codomain of A null space onto? one-to-one? X17 |X2 -X4 b. In parametric form, the solutions to the homogeneous equation AX = 0 are: = X4 X4 (free) |X5 -X6 X6 (free). In the box, write a basis for the null space of matrix A. Call the two vectors in the basis n and n2. Remember! Rows are rows and columns are columns. Note that your two basis vectors just happen to be orthogonal! (This is not always true, but the problem was designed this way.) c. Normalize your two vectors so they form an orthonormal basis u and for the null space N. Divide both vectors by their mutual 2-norm of -1 1 d. The vector ỹ = = ý + z can be written (uniquely) as the sum of a vector ý in the null space of A, and a vector ž orthogonal JDDDDD JDDDDD

Jump to level 1 Given the following bases for P2, find P, the change of coordinates matrix from B to C. C-B B = {-7x² - 15x - 12, 6x² + 2x - 7, 2x² - 4x + 3} C = {x² + x + 1, x + 1,1} P C-B Check || K î Ex: 5 Next ده 2 î ↑ ŵ ŵ 4

II. Determine if a matrix is in row-echelon form, reduced row-echelon form, or not. Click your answer to select it from the options provided below the matrix. 11) 14) [1 1 0 0 0 0 19 08 0 0 1 1 1 00018 00000 Lo 0 0 0 0 Row-Echelon Form Reduce Row-Echelon Form Not Lo 0 01 1 1 o 1. Row-Echelon Form Reduce Row-Echelon Form Not 12) Row-Echelon Form Reduce Row-Echelon Form Not 15) [1] Row-Echelon Form Reduce Row-Echelon Form Not 13) 16) Row-Echelon Form Reduce Row-Echelon Form Not 0 1 0 0 01 0 1 2 0 3 1 0 0 1 1 00015 00001 Lo 0 0 0 0 Row-Echelon Form Reduce Row-Echelon Form Not

4. The formulation of the following problem and its corresponding augumented coefficient matrix and row reduced matrix are shown below. A zoo veterinarian can purchase animal food of 4 different types, A, B, C, and D. Each food comes in the same size bag, and the number of grams of each of three nutrients in each bag are summarized in the following table. Nutrient N₁ N₂ N3 A 5 10 5 5x + 5y + 10z+5w=15,000 10x + 5y +30z+10w = 30,000 5x+15y +10z+25w=19,000 B 5 5 15 Food For one animal, the veterinarian determines that she needs to combine these bags to get 15,000 g of N₁, 30,000 g of N₂, and 19,000 g of N3. How many bags of each type of food should she order? 5 5 10 5 15,000 10 5 30 10 5 15 10 25 a. State the general solution using the parameter t. C 10 30 10 Let x = no. of bags of Food A, y = no. of bags of Food B, z = no. of bags of Food C, and w= no. of bags of Food D D 5 10 25 1 00 -3 1 0 2 0 0 1 1 30,000 →→ 0 19,000 2,200 400 200