Mathematics
Mathematics, 25.05.2021 22:50, jenhorton2011

Let U and W be subspaces of the vector space V. We know that the intersection U∩W is also a subspace of V, whereas the union U∪W is, in general,
not a subspace. In this project you will explore the sum and direct sum of subspaces,
focusing especially on their geometric interpretation in R^n.
1. Define the sum of the subspaces U and W as follows.
U+W= {u+w: u∈U, w∈W}
Prove that U+W is a subspace of V.
2. Consider the subspaces of V= R^3 listed below.
U= {(x, y,x-y): x, y ∈R}
W= {(x,0,x):x ∈R}
Z={(x, x,x): x ∈R}
Find U+W, U+Z, W+Z.
3. If U and W are subspaces of V such that V= U +W and U +W= {0}, prove that every vector in V has a unique representation of the form u+w, where u is in U and wis in W. In this case, we say that V, is the direct sum of U and W and write V=U⊕W.
(Direct sum) Which of the sums in part (2) of this project are direct sums?
4. Let V=U⊕W and suppose that {u1, u2, uk} is a basis for the
subspace U and {w1, w2, wm} is a basis for the subspace W. Prove that the set {u1,uk + w1,wm} is a basis for V.
5. Consider the subspaces of listed below.
U= {(x,0,y): x, y ∈R}
W= {(0,x, y):x, y ∈R}
Show that R^3= U+W. Is R^3 the direct sum of U and W? What are the
dimensions of U, W, U∩W, and U+W? In general, formulate a conjecture that relates the dimensions of U, W, U∩W, and U+W.
6. Can you find two 2-dimensional subspaces of R^3 whose intersection is just
the zero vector? Why or why not?

answer
Answers: 3

Other questions on the subject: Mathematics

image
Mathematics, 21.06.2019 21:30, baltazmapa629n
The length of a field is 100 yards and its width is 75 yards. if 1 inch represents 25 yards, what would be the dimensions of the field drawn on a sheet of paper?
Answers: 1
image
Mathematics, 21.06.2019 22:00, shantejahtierr63961
You're locked out of your house. the only open window is on the second floor, 25 feet above the ground. there are bushes along the edge of the house, so you will need to place the ladder 10 feet from the house. what length ladder do you need to reach the window?
Answers: 3
image
Mathematics, 21.06.2019 22:30, cordobamariana07
The difference of three and the reciprocal of a number is equal to the quotient of 20 and the number. find the number.
Answers: 2
image
Mathematics, 22.06.2019 01:00, s27511583
The weight of 46 new patients of a clinic from the year 2018 are recorded and listed below. construct a frequency distribution, using 7 classes. then, draw a histogram, a frequency polygon, and an ogive for the data, using the relative information from the frequency table for each of the graphs. describe the shape of the histogram. data set: 130 192 145 97 100 122 210 132 107 95 210 128 193 208 118 196 130 178 187 240 90 126 98 194 115 212 110 225 187 133 220 218 110 104 201 120 183 124 261 270 108 160 203 210 191 180 1) complete the frequency distribution table below (add as many rows as needed): - class limits - class boundaries - midpoint - frequency - cumulative frequency 2) histogram 3) frequency polygon 4) ogive
Answers: 1
Do you know the correct answer?
Let U and W be subspaces of the vector space V. We know that the intersection U∩W is also a subspac...

Questions in other subjects:

Konu
Mathematics, 04.04.2021 07:10
Konu
Mathematics, 04.04.2021 07:10
Konu
English, 04.04.2021 07:10