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Linear Algebra Recap: Row Space, Column Space, and Null Space

Understand the dimension, basis, and properties of row space, column space, and null space in linear algebra. Learn how to find bases and dimensions for each space. Explore the connection between rank, dimensions, and pivot columns in matrices.

emerald-mcfadden
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Linear Algebra Recap: Row Space, Column Space, and Null Space

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  1. Section 4.6 (Rank)

  2. Recap Dimension Dimension: # of vectors in basis dim Col A – number of pivot cols of A dim Nul A - # free variables in A (or number of non pivot cols of A) Note: dim Col A + dim Nul A = n

  3. Row Space DEFINITION: The set of all linear combinations of the row vectors of a matrix A, denoted by Row A

  4. Example Row A = span {r1, r2, r3} Where r1=(-1, 2, 3, 6) r2=(2, -5, -6, -12) r3=(1, -3, -3, -6) Note: Row A = Col AT

  5. Row A Observations When we use row operations to reduce Matrix A to Matrix B we are taking linear combinations of the rows of A to come up with B. Recall Row A = span {r1, r2, r3} = all linear combinations of the rows of A This leads to the following Theorem

  6. Theorem If two matrices A and B are row equivalent then, Row A = Row B If B in echelon form, the nonzero rows of B form a basis for Row A & Row B

  7. Example are row equivalent. Find a basis for row space, column space and null space of A. Also state the dimension of each. Very similar to the test question for this section

  8. Rank DEFINITION: All are equivalent dim Col A # of pivot columns of A dim Row A # nonzero rows in reduced matrix of A Rank A

  9. Rank Property Since Row A = Col AT We Have dim Row A = dim Col AT Rank A = Rank AT

  10. Rank Theorem For Amxn Rank A + dim Nul A = n OR # of pivot columns + # of non pivot columns = n

  11. Full Rank For Amxn Rank A is as large as possible OR Rank = min(m,n)

  12. Full Rank Example Given A5x7 with 5 pivot columns, is A of full rank? dim Col A = Rank A=5 = min(5,7) A is of Full Rank Pivot in each row

  13. Full Rank Observation For Amxn and m≤n(see 4.6 #26 for m>n) Full Rank Pivot in each row Columns of A Span RM There is a solution for all b in Ax=b Important in Statistics

  14. Section 4.6 Examples EXAMPLE: Suppose that a 5 x 8 matrix A has rank 5. Find dim Nul A, dim Row A and rank AT. Is Col A= R5? EXAMPLE: For a 9 x 12 matrix A, find the smallest possible value of dim Nul A.

  15. Section 4.6 Example The Rank Theorem provides us with a powerful tool for determining information about a system of equations. EXAMPLE: A scientist solves a nonhomogeneoussystem of 50 equations in 54 variables and finds that exactly 4 of the unknowns are free variables. Can the scientist be certain that any associated nonhomogeneoussystem (with the same coefficients) has a solution? Very similar to the test question for this section

  16. Additions to IMT For Amxn The columns of A form a basis of Rn Col A = Rn dim Col A = n Rank A = n Nul A = {0} dim Nul A = 0

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