Determine if the statements are true or false. Any four vectors in R^3 are linea

Question

Determine if the statements are true or false. Any four vectors in R^3 are linearly dependent. Any four vectors in R^3 span R^3. The rank of a matrix is equal to the number of pivots in its RREF. {v_1, v_2, ., v_n} is a basis for span(v_1, v_2, ., v_n). If v is an eigenvector of a matrix A, then v is an eigenvector of A + cI for all scalars c. (Her the identity matrix of the same dimension as A.) An n times n matrix A is diagonalizable if and only if it has n distinct eigenvalues. Let W be a subspace of R^n. If p is the projection of b onto W, then b - p elementof W^1.

Explanation / Answer

1. true

because only 3 vectors in R^3 may independent

hence four vectors are dependent

2. false

3.false

because the matrix rank does not agrree with the rref in pivot

4.true

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