{"content":[{"content":{"params":{"text":"

Question:<\/strong> Let \\( V \\) be the set of all lines in \\(\\mathbb{R}^2\\) with slope 1. Each such line has equation \\( y = x + d \\) for some \\(d\\in\\mathbb{R}\\). We can turn \\( V \\) into a real vector space by defining vector addition and scalar multiplication by<\/p>\n\n

\\[
\n (y=x+d_1) + (y=x+d_2) = (y=x+(d_1+d_2)) \\qquad \\qquad \\alpha(y=x + d) = (y= x + \\alpha d).
\n \\]<\/p>\n\n

What is the zero vector in \\( V \\)?<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"75b42aa5-6b48-46b1-9ff8-2dcac9ea0812"},"useSeparator":"disabled"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Solution:<\/strong> The zero vector is the line \\( y=x+0 \\). Indeed, if \\( y=x+d \\) is any line in \\( V \\), then \\[ (y=x+d) + (y=x+0) = (y=x+(d+0)) = (y=x+d). \\] So \\( y=x+0 \\) satisfies the zero vector axiom.<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"8d7ff85f-2650-4441-afae-ec58af092f91"},"title":"See Solution"}],"hTag":"h4"},"library":"H5P.Accordion 1.0","metadata":{"contentType":"Accordion","license":"U","title":"Untitled Accordion"},"subContentId":"98cddf28-460f-4de9-bd10-9fac3a08a69b"},"useSeparator":"auto"}]}

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Mary Gu