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Terms in this set (26)
linearly independent
{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3v3+... xnvn = 0 only has the trivial solution
linearly dependent
{v1, v2, v3,... vn} is linear dependent if x1v1 + x2v2 + x3v3+... xnvn = 0 has a solution
if there is a free variable...
linearly dependent
theorem 7
if the set is linearly dependent at least one of the vectors is a linear combination of the other vectors
The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution
False. Linearly independent if and only if the equation Ax=0 has ONLY the trivial solution
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S
False. If S is a linearly dependent set, then atleast one of the vectors is a linear combination of one of the other vectors in S
The columns of any 4x5 matrix are linearly dependent
True by theorem 8
If x and y are linearly independent and if {x,y,z} is linearly dependent, then z is in span {x,y}
True.
Two vectors are linearly dependent if and only if they lie on a line through the origin
True
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly dependent
False
If x and y are linearly independent, and if z is in Span {x,y}, then {x,y,z} is linearly dependent
true. If z is in the span of x and y then it is a linear combination of the two
If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector
false
Theorem 8 ***
If the number of vectors is greater than the dimension of the span than it is linearly dependent. Linearly dependent if there are more columns than rows
Theorem 9
if S contains the 0 vector, then S is linearly dependent
If v1 and v2 are in R4 and v2 is not a scalar multiple of v1 then {v1, v2} is linearly independent
False. v1 could be a zero vector