Let A = [1 2 -3 5 -4 11 4 2 2]. Find bases for Row(A), Col(A), Null(A), Row(A^T)
ID: 3032086 • Letter: L
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Let A = [1 2 -3 5 -4 11 4 2 2]. Find bases for Row(A), Col(A), Null(A), Row(A^T), Col(A^T), and Null(A^T) How many distinct subspaces are in this list? Suppose a 5 times 6 matrix A has four pivot columns. What is Nullity(A)? Is Col(A) = R^4? Explain your answers. if the nullity of a space of a 7 times 6 matrix A is 5 what is the dimension of Col(A)? If the nullity of an 8 times 5 matrix A is 2 what is the dimension of Row(A)? If A is a 4 times 3 matrix, what is the largest possible dimension of the row space of A? Explain If A is a 3 times 4 matrix what is the largest possible dimension of the row space of A? Explain If A is a 6 times 8 matrix, what is the smallest possible value for the nullity of A? If A is a 6 times 4 matrix, what is the smallest possible value for the nullity of A? Mark each answer true or false and justify your answer: The row space of A is the same as the column space of A^T. If B is the row-echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row(A) Dim Row(A) + nullity(A) = number of rows in A. Suppose A is an m times n matrix. Which of the subspaces Row(A), Col(A), Null(A), Row(A^T), Col(A^T), and Null(A^T) are in R^m and which are in R^n? How many distinct subspaces are in this list?Explanation / Answer
4. We shall reduce A to its RREF by the following row operations:
Add -5 times the 1st row to the 2nd row
Add -4 times the 1st row to the 3rd row
Multiply the 2nd row by -1/14
Add 6 times the 2nd row to the 3rd row
Multiply the 3rd row by 7/20
Add 13/7 times the 3rd row to the 2nd row
Add 3 times the 3rd row to the 1st row
Add -2 times the 2nd row to the 1st row.
Then the RREF of A is I3.This means that all the rows and columns of A are linearly independent.
Then a basis for Row(A) is {(1,2,-3), (5,-4,11),(4,2,2)}.A basis for Col(A) is {(1,5,4)T,(2,-4,2)T,(-3,11,2)T}. Further Row(AT ) and Col (AT ) are the same as Col(A) and Row(A) respectively. Further, since Null(A) is the solution set of the equation AX = 0, hence a basis for Null(A) is { (1,1,1)T }. The RREF of AT is also R3. Therefore, a basis for Null(AT) is also { (1,1,1)T }.
The distinct subspaces are Row(A), Col(A), Null(A) and Null(AT). Here, Null(A) is the same as Null(AT), but in general there are different. Thus if A is a m x n matrix, then Null(A) consists of n-vectors while Null(AT) consists of m-vectors.
5. If a 5 x 6 matrix has 4 pivot columns, it implies that it has 4 linearly independent columns so that its rank is 4. Also, its nullity is 6-4 = 2. Col(A) has a basis consisting of 4 vectors, but Col(A) is not equal to R4, as col(A) consists of 6-vectors.
6. If the nullity of a 7 x 6 matrix is 5, the dimension of Col(A) = rank of the matrix = 6 -5 = 1.
7. If the nullity of a 8 x 5 matrix is 2, the dimension of Row(A) = rank of the matrix = 5 -2 = 3.
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