Basis and Dimension (Column Space and Null Space) (Linear Algebra)
Free printable university Linear Algebra worksheet on finding bases for the column space and null space of a matrix, their dimensions, and the rank-nullity theorem.
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Linear Algebra · Math worksheet
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Linear Algebra: Basis and Dimension (Column Space and Null Space)
For each matrix, row reduce to find a basis for the column space (the pivot columns of the original matrix) and a basis for the null space (one special solution per free variable). State both dimensions and check rank + nullity = number of columns. Column vectors are written as [1; 2; 3].
- 1.For A = [1, 1, -6, 4; 1, 2, -8, 8; 2, 3, -14, 12], find a basis for the column space Col A and a basis for the null space Nul A, and state their dimensions.
- 2.For A = [1, -2, -2, 6; 2, -3, -2, 7; -2, 4, 5, -14], find a basis for the column space Col A and a basis for the null space Nul A, and state their dimensions.
- 3.For A = [1, 2, 0; -2, -3, 1; -1, -1, 1], find a basis for the column space Col A and a basis for the null space Nul A, and state their dimensions.
- 4.For A = [1, -1, 0, 1; 0, 0, 1, 2; 2, -2, -2, -2], find a basis for the column space Col A and a basis for the null space Nul A, and state their dimensions.
- 5.For A = [1, -1, 0, 1; -1, 2, 1, 0], find a basis for the column space Col A and a basis for the null space Nul A, and state their dimensions.
- 6.For A = [1, -2, -1, 3; 0, 1, 1, -2; -1, 2, 1, -3], find a basis for the column space Col A and a basis for the null space Nul A, and state their dimensions.
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