(1 point) Determine if the statements are true or false. 1. Any four vectors in
ID: 3116804 • Letter: #
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(1 point) Determine if the statements are true or false. 1. Any four vectors in R3 are linearly dependent?' 2. Any four vectors in R3 span R3 3. The rank of a matrix is equal to the number of pivots in its RREF.' ? 4(vi , v, , . . . ,%) is a basis for span(vi,v,, ,v,) . 5. If v is an eigenvector of a matrix A, then v is an eigenvector of AcI for all scalars c. (Here I denotes the identity matrix of the same dimension as A.)' 6. An n x n matrix A is diagonalizable if and only if it has n distinct eigenvalues.' 7. Let W be a subspace of R". If p is the projection of b onto W, then b peW?Explanation / Answer
1.The statement is True. The dimension of R3 is 3. Therefore, any set of 4 vectors in R3 is linearly dependent.
2.The statement is False. The set { (1,0,0),(0,1,0),(1,1,0), (1,3,0)} does not span R3.
3. The statement is True. It is one of the definitions of the rank of a matrix.
4. The statement is False. The set {v1,v2,…,vn} will be a basis for span(v1,v2,…,vn)provided the vectors v1,v2,…,vn are linearly independent and not otherwise.
5. The statement is True. If v is an eigenvector of A associated with the eigenvalue and if c is a scalar, then v is an eigenvector of A+cI with corresponding eigenvalue + c.
6. The statement is False. A nxn matrix A is diagonalizable if and only if it has n distinct and linearly independent eigenvectors. A always has n eigenvalues. Some of these may have algebraic multiplicity more than 1. This, however, may not affect the diagonalizability of A.
7. The statement is True. The vector b p is the error vector, which is perpendicular to W . Hence b-p W .
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