Checking Linear Independence in Vectors

Determining linear independence in a set of vectors involves checking if any vector can be expressed as a linear combination of others. This process involves identifying non-zero solutions for a system of linear equations, examining reduced row-echelon form matrices, and understanding the column correspondence theorem. The concept of pivot columns and non-pivot columns is crucial in determining linear independence. It is essential to distinguish between independent and dependent columns and recognize standard vectors in a matrix.

• Linear Independence
• Vectors
• Column Correspondence Theorem
• Linear Algebra

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Uploaded on Feb 28, 2025 | 0 Views

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Presentation Transcript

1. Check Independence Check Independence

2. Checking Independence Linear independent or not? A set of n vectors {a1, a2, , an} is linear dependent Given a vector set, {a1, a2, , an}, if there exists any ai that is a linear combination of other vectors matrix A Given a vector set, {a1, a2, , an}, there exists scalars x1, x2, , xn, that are not all zero, such that x1a1 + x2a2+ + xnan = 0. vector x ?? = ? have non-zero solution

3. Checking Independence Linear independent or not? A ?1 ?2 ?3 ?4 0 0 0 1 2 1 1 0 1 1 4 1 1 2 3 ?? = ? have non-zero solution or not = x1 x2x3 x4 x1 x2 x4 x3 RREF

4. Checking Independence x1 x2x3 x4 x1 x2 x4 x3 RREF dependent ?1+ 2?3= 0 ?2 ?3= 0 ?1= 2?3 ?2= ?3 ?3 ?? ???? ?4= 0 ?4= 0 ?1 ?2 ?3 ?4 2 1 1 0 ?1 ?2 ?3 ?4 setting x3 = 1 2?3 ?3 ?3 0 2 1 1 0 = = ?3 =

5. Checking Independence Linear independent or not? ! A set of n vectors {a1, a2, , an} is linear dependent Given a vector set, {a1, a2, , an}, if there exists any ai that is a linear combination of other vectors matrix A Given a vector set, {a1, a2, , an}, there exists scalars x1, x2, , xn, that are not all zero, such that x1a1 + x2a2+ + xnan = 0. vector x ?? = ? have non-zero solution

6. Column Correspondence Theorem pivot columns Leading entries linear linear independent independent The pivot columns are linear independent.

7. Column Correspondence Theorem pivot columns Leading entries a2 = 2a1 a5 = a1+a4 a6 = 5a1 3a3+2a4 r2 = 2r1 r5 = r1+r4 r6 = 5r1 3r3+2r4 The non-pivot columns are the linear combination of the previous pivot columns.

8. Independent Dependent All columns are independent The column is the linear combination of left pivot column. Every column is a pivot column If a column is not pivot Every column in RREF(A) is standard vector.

9. Independent 3X3 All columns are independent Columns are linear independent Every column is a pivot column RREF 1 0 0 0 1 0 0 0 1 Every column in RREF(A) is standard vector. Identity matrix

10. Independent 4X3 All columns are independent Columns are linear independent Every column is a pivot column RREF 1 0 0 0 0 1 0 0 0 0 1 0 ? ? Every column in RREF(A) is standard vector.

11. Independent 3X4 All columns are independent Columns are linear independent Every column is a pivot column Cannot be a pivot column RREF 1 0 0 0 1 0 0 0 1 Every column in RREF(A) is standard vector.

12. 1 2 3 1 2 3 Independent dependent The columns are dependent ( ) Dependent or Independent? dependent More than 3 vectors in R3 must be dependent. More than m vectors in Rm must be dependent.