程序代写代做代考 Math 215.01, Spring Term 1, 2021 Writing Assignment #6

Math 215.01, Spring Term 1, 2021 Writing Assignment #6
Turn your 􏰁le in on Pweb in pdf format under Writing Assign- ments”. There is no need to put your name on the document since I will grade anonymously, and Pweb will keep track of the authorship of the document.
You will be graded on your writing (use of quanti􏰁ers, state- ment and use of de􏰁nitions, and other mathematical language) as well as the validity and completeness of your mathematical arguments.
1. Find a sequence (u~1; u~2; u~3; u~4) of vectors in R3 such that whenever we omit a vector, the resulting 3 are linearly independent. You should justify why your sequence has this property.
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2. Consider the vector space R, under the usual addition and scalar multiplication (so ~0 = 0 here). Show that the only subspaces of R are f0g and R.
Hint: Let W be an arbitrary subspace of R with W 6= f0g. Weknowthat02W,sowecan􏰁xsomea2W with a 6= 0. Now explain why every element of R is in W .
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3. Let V and W be vector spaces. Suppose that T : V ! W is an injective linear transformation and that (u~1; u~2; : : : ; u~n) is a linearly independent sequence in V .
Show that (T (u~1); T (u~2); : : : ; T (u~n)) is a linearly indepen- dent sequence in W.
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4. Comment on working with partner(s): Comment on the work you and your partner(s) accomplished together and what you accomplished apart.