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In order to improve my knowledge of Linear Algebra I am reading Linear Algebra Done Right by Sheldon Axler.
In Chapter 2 under the heading Span and Linear Independence we find the following text:
"If [itex] ( v_1, v_2, ... ... v_m ) [/itex] is a list off vectors in a vector space V, then each [itex] v_j [/itex] is a linear combination of [itex] ( v_1, v_2, ... ... v_m ) [/itex].
Thus span[itex] ( v_1, v_2, ... ... v_m ) [/itex] contains each [itex] v_j [/itex].
Conversely, because because subspaces are closed under scalar multiplications and addition, every subspace V containig each [itex] v_j [/itex] must contain span[itex] ( v_1, v_2, ... ... v_m ) [/itex].
Thus the span of a list of vectors in V is the smallest subspace of V containing all the vectors in the list."
Intuitively, given that new vectors are formed by scalar multiplication and addition of other vectors that span[itex] ( v_1, v_2, ... ... v_m ) [/itex] is also the largest subspace of V containing all the vectors in the list.
Is this intuition correct? Can someone please confirm or otherwise?
Peter
In Chapter 2 under the heading Span and Linear Independence we find the following text:
"If [itex] ( v_1, v_2, ... ... v_m ) [/itex] is a list off vectors in a vector space V, then each [itex] v_j [/itex] is a linear combination of [itex] ( v_1, v_2, ... ... v_m ) [/itex].
Thus span[itex] ( v_1, v_2, ... ... v_m ) [/itex] contains each [itex] v_j [/itex].
Conversely, because because subspaces are closed under scalar multiplications and addition, every subspace V containig each [itex] v_j [/itex] must contain span[itex] ( v_1, v_2, ... ... v_m ) [/itex].
Thus the span of a list of vectors in V is the smallest subspace of V containing all the vectors in the list."
Intuitively, given that new vectors are formed by scalar multiplication and addition of other vectors that span[itex] ( v_1, v_2, ... ... v_m ) [/itex] is also the largest subspace of V containing all the vectors in the list.
Is this intuition correct? Can someone please confirm or otherwise?
Peter
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