Linear Algebra Test Questions with Answers

Linear Algebra Test Questions with Answers If A is an mxn matrix, then the columns of A are linearly independent if and only if A has .................. pivot columns - Answer-N Why is the previous question true? - Answer-The columns of a matrix A are linearly independent if and only if Ax=0 has no free variables, meaning every variable is a basic variable, that is, if and only if every column of A is a pivot column The following statement is either true or false If V1,V2,V3 are in R3 and V3 is not a linear combination of v1 v2, then {v1,v2,v3} is linearly independent - Answer-The statement is false. Take v1 and v2 to be multiples of one vector and take v3 to be not a multiple of that vector Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly dependent An indexed set of vectors {v1...vp} is said to be linearly independent if the vector equation x1v1 + x2v2 +... =0 has only a ...... - Answer-trivial solution The set{v1....vp} is said to be linearly dependent if there exist weights c1...cp that are not all what? - Answer-The weights must not all be zero such that c1v1+c2v2....=0 Determine if the vectors are linearly independent [ 5 [7 [8 0 3 9 0] -6] -36] - Answer-The vector equation has only the trivial solution so the vectors are linearly independent A system of linear equations is said to be what if it can be written an Ax=0 - Answer-homogenous