1.4 the matrix equation

The solution set of the linear system whose augmented matrix is [a_1 a_2 a_3 a_4 b] is the same as the solution set of Ax = b, if A = [a_1 a_2 a_3]

True

The 1st entry in the product Ax is a sum of products

True

The equation Ax = b is referred to as a vector equation.

False, it is a matrix equation

If the augmented matrix [A b] has a pivot position in every row, then the equation Ax=b is inconsistent

False, it may or may not be inconsistent

Suppose A is a 3x3 matrix and b is a vector in R3 w/ the property that Ax = b has a unique solution. Explain why the column of A must span R3.

If the equation has a unique solution, then the associated system of equations doesn't have any free variables. If every variable is a basic variable, then each column of A is a pivot column The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation Ax=b. Therefore the columns of A must span ℝ3.

Let A be a 3x2 matrix. Explain why the equation Ax = b can't be consistent for all b in R3. Generalize your argument to the case of an arbitrary A w/ more rows than columns

Since a 3x2 matrix only has 2 columns, matrix A can at most have 2 pivot columns and 2 pivot positions. This cannot fill all 3 rows w/ pivots so Ax=b can't be consistent for all b in R3. If A is an mxn matrix w/ m>n then A can have at most n pivot positions which isn't enough to fill all m rows, so Ax = b can't be consistent for all b in R3. "When written in reduced echelon​ form, any m×n matrix will have at least one row of all zeros. When solving Ax=b​, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side."

The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row.

The statement is false. The equation Ax=b may or may not be consistent if the augmented matrix [A b] has a pivot position in every row. If one of the pivot positions occurs in the column that represents b​, then the equation is not consistent. If all of the pivot positions occur in columns that represent​ A, then the equation is consistent. ​However, the equation Ax=b is consistent if the matrix A has a pivot position in every row.

A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.

True

Every matrix equation Ax=b corresponds to a vector equation with the same solution set.

True

If the columns of an mxn matrix A span Rm, then the equation Ax=b is consistent for each b in Rm.

True

If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns of A.

True

Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.

True, Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x​, where A is a matrix of the coefficients of the system of vectors.

If A is an mxn matrix whose columns span Rm, then the equation Ax=b is consistent for some b in Rm

True. Consult the following theorem. Let A be an m×n matrix. Then the following statements are logically equivalent. That​ is, for a particular​ A, either they are all true statements or they are all false. a. For each b in ℝm​, the equation Ax=b has a solution. b. Each b in ℝm is a linear combination of the columns of A. c. The columns of A span ℝm. d. A has a pivot position in every row. If the columns of A span ℝm​, then statement c is true.​ Therefore, statement a is true. If the equation Ax=b has a solution for each b in ​, then the equation Ax=b is consistent for each b in ℝm.

If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row

True. Let A be an m×n matrix. Then the following statements are logically equivalent. That​ is, for a particular​ A, either they are all true statements or they are all false. a. For each b in ℝm​, the equation Ax=b has a solution. b. Each b in ℝm is a linear combination of the columns of A. c. The columns of A span ℝm. d. A has a pivot position in every row. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm​, then the equation Ax=b has no solution for some b in ℝm. Statement a is false.​ Therefore, statement d is also false. This means that A cannot have a pivot position in every row.

Could a set of 3 vectors in R4 span all of R4? What about n vectors in Rm when n < m?

a) A set of 3 vectors in R4 can't span R4 because matrix A would have 4 rows. There needs to be at least 4 columns (one for each pivot). Since it doesn't have a pivot in each row, the column do NOT span R4. b) A set of n vectors in Rm can't span Rm when n < m. "Generally, if A is an m×n matrix with m>​n, then A can have at most n pivot​ positions, which is not enough to fill all m rows."

Suppose A is a 4x3 matrix and b is a vector in R4 w/ the property that Ax = b has a unique solution. What can you say about the reduced echelon form of A?

Since it is unique, there are no free variables in A, so each column of A is a pivot column because every variable is a basic variable.

Let A be a 3x4 matrix, let y_1 and y_2 be vectors in R3, and let w = y_1 + y_2. Suppose y_1 = Ax_1, y_2 = Ax_2 for some vectors x_1 and x_2 in R2. What fact allows you to conclude that the system Ax = w is consistent? (x_1 and x_2 denote vectors, not scalar entries in vectors)

w = Ax_1 + Ax_2 = A(x_1 + x_2) so vector x = x_1 + x_ 2 is a solution of w = Ax

Let A be a 5x3 matrix, let y be a vector in R3, and let z be a vector in R5. Suppose Ay = z. What fact allows you to conclude that the system Ax = 5z is consistent?

y and z satisfy Ay = z, so 5z = 5Ay (see theorem 5) which shows that 5y is a solution of Ax = 5z, so Ax = 5z is consistent.