Describe the column spaces (lines or planes) of these particular matrices:
A equals |1 2 0 0 0 0 B equals |1 0 0 2 0 0 C equals|1 0 2 0 0 0
Solution
A is spanned by the single vector {[1]} because v2 can be written as {2[v1]} A set spanned by a single vector is a line. B forms a linearly independent set (Ax = 0 has only trivial solutions). Therefore the basis of B is spanned by set the {v1,v2}. For clarity, lets say Vector 1 horizontally is [x y z] and Vector 2 is [x y z] The matrix spans the region between c1[v1] + c2[v2] = v So we have c1(x) + c2(y) = v.. Picture 2 vectors through the origin, 1 going +1 in the x direction, and 1 going +2 in the y direction. These vectors form a parallelogram in which their multiples are contained. i.e. A plane. C forms a linearly independent set (Ax = 0) has only trivial solutions. However, every line formed by c1[v1] + c2[v2] = v can also be written as c1[v1] = v Therefore the set is spanned by the single vector {v1}, and is therefore a line.
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Describe the column spaces (lines or planes) of these particular matri.docx
1. Describe the column spaces (lines or planes) of these particular matrices:
A equals |1 2 0 0 0 0 B equals |1 0 0 2 0 0 C equals|1 0 2 0 0 0
Solution
A is spanned by the single vector {[1]} because v2 can be written as {2[v1]} A set spanned by a
single vector is a line. B forms a linearly independent set (Ax = 0 has only trivial solutions).
Therefore the basis of B is spanned by set the {v1,v2}. For clarity, lets say Vector 1 horizontally
is [x y z] and Vector 2 is [x y z] The matrix spans the region between c1[v1] + c2[v2] = v So we
have c1(x) + c2(y) = v.. Picture 2 vectors through the origin, 1 going +1 in the x direction, and 1
going +2 in the y direction. These vectors form a parallelogram in which their multiples are
contained. i.e. A plane. C forms a linearly independent set (Ax = 0) has only trivial solutions.
However, every line formed by c1[v1] + c2[v2] = v can also be written as c1[v1] = v Therefore
the set is spanned by the single vector {v1}, and is therefore a line.
![Describe the column spaces (lines or planes) of these particular matrices:
A equals |1 2 0 0 0 0 B equals |1 0 0 2 0 0 C equals|1 0 2 0 0 0
Solution
A is spanned by the single vector {[1]} because v2 can be written as {2[v1]} A set spanned by a
single vector is a line. B forms a linearly independent set (Ax = 0 has only trivial solutions).
Therefore the basis of B is spanned by set the {v1,v2}. For clarity, lets say Vector 1 horizontally
is [x y z] and Vector 2 is [x y z] The matrix spans the region between c1[v1] + c2[v2] = v So we
have c1(x) + c2(y) = v.. Picture 2 vectors through the origin, 1 going +1 in the x direction, and 1
going +2 in the y direction. These vectors form a parallelogram in which their multiples are
contained. i.e. A plane. C forms a linearly independent set (Ax = 0) has only trivial solutions.
However, every line formed by c1[v1] + c2[v2] = v can also be written as c1[v1] = v Therefore
the set is spanned by the single vector {v1}, and is therefore a line.](https://image.slidesharecdn.com/describethecolumnspaceslinesorplanesoftheseparticularmatri-230131021742-32dd8527/85/Describe-the-column-spaces-lines-or-planes-of-these-particular-matri-docx-1-320.jpg)
