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.