If 𝑆 = {𝑣1 , 𝑣2, . . . , 𝑣𝑛 } is a set of vectors in a finite dimensional vector space 𝑉 , then 𝑆 is called a basis for 𝑉 if:
Question
If 𝑆 = {𝑣1 , 𝑣2, . . . , 𝑣𝑛 } is a set of vectors in a finite dimensional vector space 𝑉 , then 𝑆 is called a basis for 𝑉 if:
Solution
S is called a basis for V if it satisfies two conditions:
-
S is a linearly independent set: This means that no vector in the set S can be written as a linear combination of the other vectors in the set. In other words, there are no redundant vectors in S.
-
S spans V: This means that every vector in V can be written as a linear combination of the vectors in S. In other words, S is a generating set for V.
If both these conditions are met, then S is a basis for the vector space V.
Similar Questions
Consider the matrixA=⎡⎣⎢212424−12042−5⎤⎦⎥.Find a basis for the column space of A. A.⎧⎩⎨⎪⎪⎡⎣⎢212⎤⎦⎥,⎡⎣⎢424⎤⎦⎥,⎡⎣⎢−120⎤⎦⎥⎫⎭⎬⎪⎪ B.⎧⎩⎨⎪⎪⎡⎣⎢212⎤⎦⎥,⎡⎣⎢−120⎤⎦⎥,⎡⎣⎢42−5⎤⎦⎥⎫⎭⎬⎪⎪ C.⎧⎩⎨⎪⎪⎡⎣⎢100⎤⎦⎥,⎡⎣⎢010⎤⎦⎥,⎡⎣⎢001⎤⎦⎥⎫⎭⎬⎪⎪ D.⎧⎩⎨⎪⎪⎡⎣⎢200⎤⎦⎥,⎡⎣⎢−110⎤⎦⎥,⎡⎣⎢401⎤⎦⎥⎫⎭⎬⎪⎪
Find a basis for the column space ofA=⎡⎣⎢⎢−43820−4−2−34010⎤⎦⎥⎥.
Which of the following sets spans the vector spaceR3? Select all that apply. A. ⎧⎩⎨⎪⎪⎡⎣⎢211⎤⎦⎥,⎡⎣⎢301⎤⎦⎥,⎡⎣⎢512⎤⎦⎥,⎡⎣⎢602⎤⎦⎥⎫⎭⎬⎪⎪ B. ⎧⎩⎨⎪⎪⎡⎣⎢110⎤⎦⎥,⎡⎣⎢300⎤⎦⎥⎫⎭⎬⎪⎪ C. ⎧⎩⎨⎪⎪⎡⎣⎢201⎤⎦⎥,⎡⎣⎢110⎤⎦⎥,⎡⎣⎢001⎤⎦⎥⎫⎭⎬⎪⎪ D. ⎧⎩⎨⎪⎪⎡⎣⎢100⎤⎦⎥,⎡⎣⎢012⎤⎦⎥,⎡⎣⎢003⎤⎦⎥⎫⎭⎬⎪⎪
onsider a linearly independent spanning set v1, . . . , vn. Consider a subset v1, . . . , vm, wherem < n. Is v1, . . . , vm spanning or not spanning? In your answer, you may use definitions andproperties of linearly independent sets and spanning sets, but not the concepts of a basis anddimension
Find the basis for the kernel of the matrixA=⎛⎝⎜11−12−105−41001⎞⎠⎟A. {⎛⎝⎜⎜⎜1−311⎞⎠⎟⎟⎟} B. {⎛⎝⎜⎜⎜1−310⎞⎠⎟⎟⎟}C. {⎛⎝⎜⎜⎜1310⎞⎠⎟⎟⎟} D. {⎛⎝⎜⎜⎜−1310⎞⎠⎟⎟⎟}
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.