1.u + v is in V2.u + v = v + u3.(u + v) + w = u + (v + w)4.There is a 0 vector in V such that u + 0 = ua.Note: the zero vector is different in each vector space. For example, a vector V can be in R3 or R2. they aretwo different zero vectors5.u + (-u) = 06.cu is in V7.c(u + v) = cu + cv8.(c + d) u = cu + du9.c(du) = (cd)u10.1u = u-Rn, where n >= 1, is a good EXAMPLE of vector spaces. Geometrically think about R3 to help you visualize the otherconcepts of vector spaces.-Polynomials are also vector spaces because they follow the axioms. Multiplying a polynomial by a scalar gives you apolynomial. Adding two polynomials together gives you a polynomial.-So far we have extended many of the properties of R2 and R3 to Rn . We don’t have to stop there. By taking the mostfundamental properties of these “natural” sets of vectors, we can define a general vector space. Any set of objectsequipped with an addition and scalar multiplication which satisfies the eight vector space axioms qualifies as a vectorspace.-Basically, we know about the geometric Rn space.-But anything can be a vector space (doesn’t need to be explicitly a coordinate system) as long as it is closed underaddition and multiplication.SUBSPACESTo check if a space H is a subspace of vector space V, you have to check for 3 of the 10 above axioms. The rest are automaticallysatisfied.a.The zero vector of V is in Hb.H is closed under vector additionc.H is closed under multiplication by scalars.-Every vector space is a subspace of some larger vector space.

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-Every subspace is a vector space-Sayingsubspace of Vlets you know that V is the lager vector space.A SUBSPACE SPANNED BY A SET-One of the most common ways of describing a subspace is by spans.-Recall: a span is the space of the possible linear combinations of vectorsTheorem 1:If v1...vp are in a vector space V, then Span{v1...vp} is a subspace of V.-We call Span{v1...vp} the subspace spanned (or generated) by {v1….vp}-This is because a span is basically scaling and adding the vectors together. And a vector space is closed under addition andmultiplication by scalar. So naturally, the span is also in the vector space.Questions I may be asked-For what values of h will y be in the subspace of R3 spanned by v1, v2, v3. You will be given v1...v3. And y will have h as amissing variable. So, basically find the value of h so that the vector y is in the subspace of R3-To do so, you are checking to see if y is in the span of v1...v3.-Solve this like you solved other problems. Check if y is a linear combination of v1...v3.-So put v1...v3 in a matrix and the y vector as the augmented last column.-Solve and see what you need h to be.If asked how you know that a span of a certain vectors is a subspace of Rn, say →-You know it is a subspace of Rn because of Theorem 1-This states that a span of vectors in Rn is a subspace of that Rn4.2 Null Spaces, Column Spaces, and Linear TransformationsNull and Column spaces are both vector subspaces that are associated with matrices. (they mark the solution sets in V that solve yourparticular matrices)-

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Term

Spring

Professor

Chorin