Created on 2024-06-20Asked by Avery Campbell (Solvelet student)
Find the null space and column space of the matrix: A=⎝⎛147258369⎠⎞.
Solution
To find the null space and column space of the matrix A=⎝⎛147258369⎠⎞, follow these steps: 1. **Null Space**: - Solve Ax=0. Perform row reduction on A: ⎝⎛147258369⎠⎞R2←R2−4R1⎝⎛1072−383−69⎠⎞R3←R3−7R1⎝⎛1002−3−63−6−12⎠⎞R3←R3−2R2⎝⎛1002−303−60⎠⎞. The system reduces to: {x1+2x2+3x3=0−3x2−6x3=0. From the second equation, x2=−2x3. Substituting into the first equation: x1+2(−2x3)+3x3=0⟹x1−x3=0⟹x1=x3. Therefore, the null space is: Null(A)=span⎩⎨⎧⎝⎛1−21⎠⎞⎭⎬⎫. 2. **Column Space**: - The pivot columns in the row-reduced form of A are the first and second columns. - Therefore, the column space is spanned by the corresponding columns of the original matrix: Col(A)=span⎩⎨⎧⎝⎛147⎠⎞,⎝⎛258⎠⎞⎭⎬⎫.Solved on Solvelet with Basic AI Model
Some of the related questions asked by Sophia White on Solvelet
DefinitionThe Null Space of a matrix A is the set of all vectors x in R^m (m x 1) such that Ax=0. The column space is nothing else than what his name suggests. It is the span of all of the columns of A. These are fundamental ideas to solve linear systems and to understand properties of matrices. For instance: If A=(1326) then NULLSPACE = (−21) and COLSPACE = (13).
